Difference between revisions of "1969 AHSME Problems/Problem 11"

Latest revision as of 03:13, 7 June 2018

Problem

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$ -plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals:

$\text{(A) } -\tfrac{3}{5}\quad \text{(B) } -\tfrac{2}{5}\quad \text{(C) } -\tfrac{1}{5}\quad \text{(D) } \tfrac{1}{5}\quad \text{(E) either }-\tfrac{1}{5}\text{ or} \tfrac{1}{5}.$

Solution

$[asy] import graph; size(7.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.2,xmax=6.2,ymin=-4.2,ymax=4.2; pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); Label laxis; laxis.p=fontsize(10); xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); dot((-1,-2)); label("P",(-1,-2),NW); dot((4,2)); label("Q",(4,2),NW); draw((-1,-2)--(4,2),dotted); dot((1,-0.4)); label("R",(1,-0.4),SE); [/asy]$ By the Triangle Inequality, $PR + QR \ge PR$ , and equality holds if $R$ is on $PQ$ . The equation of the line with $P$ and $Q$ is $y = \frac{4}{5}x - \frac{6}{5}$ , so point $R$ is $(1,-\frac{2}{5})$ . Thus, $m = \boxed{\textbf{(B) } -\frac{2}{5}}$ .

1969 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
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Difference between revisions of "1969 AHSME Problems/Problem 11"

Latest revision as of 03:13, 7 June 2018

Problem

Solution

See also

@@ Line 10: / Line 10: @@
 == Solution ==
-<math>\fbox{B}</math>
+<asy>
+import graph; size(7.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black;
+real xmin=-2.2,xmax=6.2,ymin=-4.2,ymax=4.2;
+pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0);
+/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
+for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
+Label laxis; laxis.p=fontsize(10);
+xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
+clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
+dot((-1,-2));
+label("P",(-1,-2),NW);
+dot((4,2));
+label("Q",(4,2),NW);
+draw((-1,-2)--(4,2),dotted);
+dot((1,-0.4));
+label("R",(1,-0.4),SE);
+</asy>
+By the [[Triangle Inequality]], <math>PR + QR \ge PR</math>, and equality holds if <math>R</math> is on <math>PQ</math>.  The equation of the line with <math>P</math> and <math>Q</math> is <math>y = \frac{4}{5}x - \frac{6}{5}</math>, so point <math>R</math> is <math>(1,-\frac{2}{5})</math>.  Thus, <math>m = \boxed{\textbf{(B) } -\frac{2}{5}}</math>.
 == See also ==
-{{AHSME box|year=1969|num-b=10|num-a=12}}
+{{AHSME 35p box|year=1969|num-b=10|num-a=12}}
 [[Category: Introductory Geometry Problems]]
 {{MAA Notice}}